The resolvent algebra: A new approach to canonical quantum systems. (English) Zbl 1148.46032

Buchholz, Detlev; Grundling, Hendrik

J. Funct. Anal. 254, No. 11, 2725-2779 (2008).

Summary: The standard \(C^{*}\)-algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. Here a novel \(C^{*}\)-algebra of the canonical commutation relations is presented which does not suffer from such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting \(C^{*}\)-algebra, the resolvent algebra, is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems, as we demonstrate by several examples.

MSC:

46L05	General theory of \(C^*\)-algebras
46L60	Applications of selfadjoint operator algebras to physics
81T05	Axiomatic quantum field theory; operator algebras

Keywords:

resolvent algebra; canonical commutation relations; quantum field theory; \(C^{*}\)-algebra; Weyl algebra; representation; Bosonic field

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